Let's go through each part in detail using the vertex form of a quadratic function. The vertex form is:

$f(x) = a(x - h)^2 + k$

where:

• $(h, k)$ is the vertex of the parabola,
• $a$ is the leading coefficient that determines the "width" and direction of the parabola.

Part 1: Determining the Sign of the Leading Coefficient

Since the parabola opens upward (the values increase as $x$ moves away from the vertex), the leading coefficient $a$ must be positive.

Answer: Positive.

Part 2: Values of $h$ and $k$

From the problem, we know that the vertex is $(2, 8)$. Thus:

• $h = 2$
• $k = 8$

Answer: $h = 2$ and $k = 8$.

Part 3: Value of the Leading Coefficient $a$

We know that the function passes through the point $(3, 13)$. Substitute $(x, y) = (3, 13)$ and $(h, k) = (2, 8)$ into the vertex form equation:

$13 = a(3 - 2)^2 + 8$

Simplify:

$13 = a(1)^2 + 8$

$13 = a + 8$

Subtract 8 from both sides:

$a = 5$

Answer: $a = 5$.

Part 4: Equation of the Function in Vertex Form

Now that we know $a = 5$, $h = 2$, and $k = 8$, we can write the equation as:

$f(x) = 5(x - 2)^2 + 8$

Answer: $f(x) = 5(x - 2)^2 + 8$.

Would you like further explanation on any step, or is there a specific part you'd like to explore more?

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Here are five follow-up questions to deepen understanding:

1. What would happen to the parabola if $a$ were negative instead of positive?
2. How can we find the axis of symmetry for a quadratic function in vertex form?
3. How do changes in $h$ and $k$ affect the position of the vertex on the coordinate plane?
4. How can you convert this function from vertex form to standard form $ax^2 + bx + c$?
5. What does the vertex tell us about the maximum or minimum of the function?

Tip: When identifying the direction of a parabola (upward or downward), always look at the sign of the leading coefficient $a$. Positive $a$ means it opens up, and negative $a$ means it opens down.